To express vector x as a linear combination of vectors a and b, we want to find scalars $\alpha$ and $\beta$ such that:

$x = \alpha a + \beta b$

From the image:

• The vector a points from the origin to approximately (2, -1).
• The vector b points from the origin to approximately (2, 2).
• The vector x points from the origin to approximately (-7, -5).

Now we need to solve the equation:

$\begin{pmatrix} -7 \\ -5 \end{pmatrix} = \alpha \begin{pmatrix} 2 \\ -1 \end{pmatrix} + \beta \begin{pmatrix} 2 \\ 2 \end{pmatrix}$

This leads to the system of linear equations:

$-7 = 2\alpha + 2\beta$ $-5 = -\alpha + 2\beta$

Let's solve this system to find $\alpha$ and $\beta$.The solution to the system of equations is:

$\alpha = -\frac{2}{3} \quad \text{and} \quad \beta = -\frac{17}{6}$

Rounding these to the nearest tenth, we get:

$\alpha \approx -0.7 \quad \text{and} \quad \beta \approx -2.8$

Thus, the vector x can be written as:

$x \approx -0.7a - 2.8b$

Would you like more details or have any questions?

Here are some related questions you could explore:

1. What is the geometric meaning of expressing one vector as a linear combination of others?
2. How do we know if vectors a and b are linearly independent?
3. What happens if a and b were parallel vectors?
4. How would this method apply in three dimensions?
5. Can we express x as a linear combination if a and b were in different quadrants?

Tip: Always check if vectors are linearly independent before expressing one as a combination of others, as this ensures a unique solution.